4 ms·

a complex number is a point in the plane (in the same way as a real number is a point in a straight line)

by enriquto 8y ago

empiricus 8y ago
What does it mean to multiply two points in the plane? You still need an additional rule for that.
- enriquto 8y ago
  > What does it mean to multiply two points in the plane? You still need an additional rule for that. yes, you do. I was answering the question of what complex numbers are. There are many things that you can do with them, that you need to define separately.
  - ttoinou 8y ago
    a 2D point is not enough to define complex numbers
    - enriquto 8y ago
      > a 2D point is not enough to define complex numbers alright. Yet, one 2d point is certainly enough to define one complex number, which is what I meant. The data is the same, but they have different methods.
      - navane 8y ago
        I struggle with this a lot, too. Maybe "complex numbers" is a misnomer, and we should call them "complex operations", or "complex methods". I have a hunch that for math people complex numbers are tied really close to the operations you perform with them. But to me, the lay man, the numbers are just data. From reading around on HN I infer that there are certain things that are grouped together because they share certain actions. Groups, sets, monads. However, if you were to encounter the number (1, 5) in the wild, you wouldnt know if it was a complex number or just a 2 dimensional vector until it is operated upon.
      - ttoinou 8y ago
        The OP was looking for "explanation of complex numbers" You won't explain anything by talking about 2D points only But you can if you specify only how they behave with addition and multiplication (see my others posts)
        enriquto 8y ago
        > The OP was looking for "explanation of complex numbers" > You won't explain anything by talking about 2D points only The OP was complaining about the explanation that complex numbers "have a real and an imaginary part", because it mean nothing to them. It is alright, then, to clarify that this just means that they are points in the plane, and some operations will be defined on these points.
        ttoinou 8y ago
        Yes
  - darkscape 8y ago
    Hmm, I tend to think that what one can do with some objects is an essential part of what they are. Two-dimensional vectors can just as well be thought of as points in the plane. What sets complex numbers apart is eg that we define complex multiplication of them. A more general example: a group is not just some set G, but that set with a binary operation satisfying certain axioms.
dragonwriter 8y ago
> a complex number is a point in the plane (in the same way as a real number is a point in a straight line) More accurately: a complex number is not a point in the plane (in the same way as a real number is not a point in a straight line) Given a line and an arbitrary selection of a 0-point and positive direction, you can map between real numbers and points on the line, and you can do a similar thing with complex numbers and points on a plane (which requires separate selection of positive directions for the real and complex axes.) But real numbers aren't points on a line, and complex numbers aren't points on a plane.
adrian_b 8y ago
While it is true that 2 real numbers can determine both a complex number and a point in plane and it is also true that 1 real number can determine both a real number and a point in a straight line, these are different mathematical things. The correct view is that the real numbers, also known as scalars, are quotients of 1-dimensional vectors, while the complex numbers are quotients of 2-dimensional vectors. This means that given two 1-dimensional vectors, the second being non-null, there is always a real number by which you can multiply the second vector to obtain the first vector. The same for two 2-dimensional vectors and a complex number. In this case the magnitude of the complex number changes the magnitude of the vector, while the phase rotates the vector. When you understand this fact about complex numbers, than it becomes obvious that the imaginary unit is not imaginary but just a rotation with a right angle and it is trivial that its square, i.e. a rotation with 180 degrees is equivalent with a multiplication by -1. The point are a third, different kind of mathematical entities, distinct from both real & complex numbers and from vectors. In fact points (i.e. members of 1-dimensional, 2-dimensional and so on affine spaces) are the primitive objects. From points you can define the vectors as differences of points (i,e, translations). Then from vectors you can define real numbers, complex numbers, quaternions and higher-dimensional matrices as quotients of vectors. (Such definitions define e.g. a 2-dimensional vector as a class of equivalence of pairs of points in plane that are transformed into each other by a translation, and a complex number as a class of equivalence of pairs of 2-dimensional vectors which are transformed into each other by a proportional change in magnitude and a rotation with a fixed angle.) Obviously, my explanation here is very simplified, but it is useful to be aware that e.g. a point in plane, a 2-dimensional vector and a complex number are 3 very different kinds of mathematical objects, even if all 3 are determined by a pair of real numbers. They are very different because the set of operations that can be applied to each is different. For example only the complex numbers are members of a field. You can add 2-dimensional vectors (i.e. compose 2 translations), but you cannot add 2 points from a plane.
- ttoinou 8y ago
  How do you define a 1D vector then ?
  - 8y ago

a complex number is a point in the plane (in the same way as a real number is a point in a straight line)

empiricus 8y ago

What does it mean to multiply two points in the plane? You still need an additional rule for that.

enriquto 8y ago
> What does it mean to multiply two points in the plane? You still need an additional rule for that. yes, you do. I was answering the question of what complex numbers are. There are many things that you can do with them, that you need to define separately.
- ttoinou 8y ago
  a 2D point is not enough to define complex numbers
  - enriquto 8y ago
    > a 2D point is not enough to define complex numbers alright. Yet, one 2d point is certainly enough to define one complex number, which is what I meant. The data is the same, but they have different methods.
    - navane 8y ago
      I struggle with this a lot, too. Maybe "complex numbers" is a misnomer, and we should call them "complex operations", or "complex methods". I have a hunch that for math people complex numbers are tied really close to the operations you perform with them. But to me, the lay man, the numbers are just data. From reading around on HN I infer that there are certain things that are grouped together because they share certain actions. Groups, sets, monads. However, if you were to encounter the number (1, 5) in the wild, you wouldnt know if it was a complex number or just a 2 dimensional vector until it is operated upon.
    - ttoinou 8y ago
      The OP was looking for "explanation of complex numbers" You won't explain anything by talking about 2D points only But you can if you specify only how they behave with addition and multiplication (see my others posts)
      - enriquto 8y ago
        > The OP was looking for "explanation of complex numbers" > You won't explain anything by talking about 2D points only The OP was complaining about the explanation that complex numbers "have a real and an imaginary part", because it mean nothing to them. It is alright, then, to clarify that this just means that they are points in the plane, and some operations will be defined on these points.
        ttoinou 8y ago
        Yes
- darkscape 8y ago
  Hmm, I tend to think that what one can do with some objects is an essential part of what they are. Two-dimensional vectors can just as well be thought of as points in the plane. What sets complex numbers apart is eg that we define complex multiplication of them. A more general example: a group is not just some set G, but that set with a binary operation satisfying certain axioms.

dragonwriter 8y ago

> a complex number is a point in the plane (in the same way as a real number is a point in a straight line) More accurately: a complex number is not a point in the plane (in the same way as a real number is not a point in a straight line) Given a line and an arbitrary selection of a 0-point and positive direction, you can map between real numbers and points on the line, and you can do a similar thing with complex numbers and points on a plane (which requires separate selection of positive directions for the real and complex axes.) But real numbers aren't points on a line, and complex numbers aren't points on a plane.

adrian_b 8y ago

While it is true that 2 real numbers can determine both a complex number and a point in plane and it is also true that 1 real number can determine both a real number and a point in a straight line, these are different mathematical things. The correct view is that the real numbers, also known as scalars, are quotients of 1-dimensional vectors, while the complex numbers are quotients of 2-dimensional vectors. This means that given two 1-dimensional vectors, the second being non-null, there is always a real number by which you can multiply the second vector to obtain the first vector. The same for two 2-dimensional vectors and a complex number. In this case the magnitude of the complex number changes the magnitude of the vector, while the phase rotates the vector. When you understand this fact about complex numbers, than it becomes obvious that the imaginary unit is not imaginary but just a rotation with a right angle and it is trivial that its square, i.e. a rotation with 180 degrees is equivalent with a multiplication by -1. The point are a third, different kind of mathematical entities, distinct from both real & complex numbers and from vectors. In fact points (i.e. members of 1-dimensional, 2-dimensional and so on affine spaces) are the primitive objects. From points you can define the vectors as differences of points (i,e, translations). Then from vectors you can define real numbers, complex numbers, quaternions and higher-dimensional matrices as quotients of vectors. (Such definitions define e.g. a 2-dimensional vector as a class of equivalence of pairs of points in plane that are transformed into each other by a translation, and a complex number as a class of equivalence of pairs of 2-dimensional vectors which are transformed into each other by a proportional change in magnitude and a rotation with a fixed angle.) Obviously, my explanation here is very simplified, but it is useful to be aware that e.g. a point in plane, a 2-dimensional vector and a complex number are 3 very different kinds of mathematical objects, even if all 3 are determined by a pair of real numbers. They are very different because the set of operations that can be applied to each is different. For example only the complex numbers are members of a field. You can add 2-dimensional vectors (i.e. compose 2 translations), but you cannot add 2 points from a plane.

ttoinou 8y ago
How do you define a 1D vector then ?
- 8y ago